Place potatoes in a medium pot and add cold water to cover by 1"; season with salt and bring to a boil. Reduce heat and simmer until fork-tender, 15–20 minutes. Drain, reserving 1/4 cup cooking liquid. Return potatoes to pot; add 2 Tbsp. cooking liquid and 2 Tbsp. oil and lightly mash. Season with salt. Keep potatoes warm over low heat until ready to serve.

Combine leeks, thyme, garlic, and milk in a medium wide saucepan. Season generously with salt and bring to a simmer over medium heat. Cook until leeks are just beginning to soften, 6–8 minutes. Gently slide cod into infused milk and poach until flesh is cooked through and beginning to flake, 7–9 minutes (adjust heat as needed to maintain a low simmer; thicker pieces of fish will take longer to cook).

Divide warm potatoes, leeks, and cod among shallow bowls. Spoon some poaching liquid over each. Drizzle with oil; season with pepper.